Gelfand-Kirillov Dimension And Reducibility Of Scalar Generalized Verma Modules For Classical Lie Algebras

Let g be a classical Lie algebra and p be a maximal parabolic subalgebra.Let M be a generalized Verma module induced from a onedimensional representation of p.Such M is called a scalar generalized Verma module.Its simple quotient L is the highest weight module.In this paper,we will determine the reducibility of such scalar generalized Verma modules by computing the Gelfand-Kirillov(GK)dimension of the corresponding highest weight modules and the explicit formula of Gelfand-Kirillov dimension for scalar generalized Verma modules.The specific methods to determine the reducibility of scalar generalized Verma modules are as follows:We know that the set of reducible points of a scalar generalized Verma module MI(λ)(with λ=zξ and ξ=ξp being the fundamental weight corresponding to the simple root αp)contains the following diagram:——● ● ●……●……a where the reducible points starting from z=a ∈R are equally spaced at an interval of length 1 and are like the form a+Z≥0,the point a will be called the first reducible point of MI(λ).We only need to find the first reducible points of the scalar generalized Verma module MI(zξ).Note that from[15,Lemma 1.6],there exist finitely many first reducible points for a given scalar generalized Verma module We will also prove this result by using the GK dimension.By the criteria of reducibility and Young tableaux,we give the set of reducible points of the scalar generalized Verma module,we also give the GK dimension formulas of the reducible points,see[Theorem 3.4,Theorem 3.6,Theorem 3.14,Theorem 3.16,Theorems 4.1-4.11]for more details.This thesis contains the following five chapters.In Chapter 1,we introduce the background and progress in the reduciblity of such scalar generalized Verma modules and list the main results of this thesis.In Chapter 2,we give the form of the maximal parabolic subalgebra and corresponding dimension of u.We will also define the GelfandKirillov dimension and corresponding formulas,and this chapter is the basis for the results of this thesis.In Chapter 3,we describe the set of reducible points of the scalar generalized Verma modules.In Chapter 4,we determine the explicit formula of GK dimension for scalar-type highest weight modules.In Chapter 5,we mainly write about the reason why this thesis does not discuss the reducibility of scalar generalized Verma modules for exceptional Lie algebras.